Epicycloid

In geometry, an epicycloid or hypercycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is a particular kind of roulette.

The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).

EquationsEdit

If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by either:

 
 

or:

 
 

(Assuming the initial point lies on the larger circle.)

If k is a positive integer, then the curve is closed, and has k cusps (i.e., sharp corners).

If k is a rational number, say k = p / q expressed as irreducible fraction, then the curve has p cusps.

To close the curve and
complete the 1st repeating pattern :
θ = 0 to q rotations
α = 0 to p rotations
total rotations of outer rolling circle = p + q rotations

Count the animation rotations to see p and q .

If k is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius R + 2r.

The distance OP from (x=0,y=0) origin to (the point   on the small circle) varies up and down as

R <= OP <= (R + 2r)

R = radius of large circle and

2r = diameter of small circle

The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid, two cusps is a nephroid.

An epicycloid and its evolute are similar.[1]

ProofEdit

 
sketch for proof

We assume that the position of   is what we want to solve,   is the radian from the tangential point to the moving point  , and   is the radian from the starting point to the tangential point.

Since there is no sliding between the two cycles, then we have that

 

By the definition of radian (which is the rate arc over radius), then we have that

 

From these two conditions, we get the identity

 

By calculating, we get the relation between   and  , which is

 

From the figure, we see the position of the point   on the small circle clearly.

 
 

See alsoEdit

ReferencesEdit

  • J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 161, 168–170, 175. ISBN 978-0-486-60288-2.

External linksEdit